Question

find the graph of this function as the value of n increases, starting from n = 1.
f(n) = (\frac{12}{17}+\frac{13}{17}i)^n
remember: |a + bi| = \sqrt{a^{2}+b^{2}}

Explanation:

Step1: Find the modulus of the complex - number

For the complex number $z=\frac{12}{17}+\frac{13}{17}i$, using the formula $|a + bi|=\sqrt{a^{2}+b^{2}}$, we have $|z|=\sqrt{(\frac{12}{17})^{2}+(\frac{13}{17})^{2}}=\sqrt{\frac{144 + 169}{289}}=\sqrt{\frac{313}{289}}\approx1$.

Step2: Recall De - Moivre's theorem

De - Moivre's theorem states that for a complex number $z = r(\cos\theta+i\sin\theta)$ and a positive integer $n$, $z^{n}=r^{n}(\cos(n\theta)+i\sin(n\theta))$. Since $r\approx1$, as $n$ increases, the points $z^{n}$ lie on the unit - circle (because $r^{n}\approx1^{n} = 1$ for all positive integers $n$). The angle $\theta$ of the complex number $z=\frac{12}{17}+\frac{13}{17}i$ is $\theta=\arctan(\frac{13}{12})$. As $n$ increases, the points $f(n)=z^{n}$ are evenly spaced around the unit - circle.

Answer:

The points $f(n)$ lie on the unit - circle and are evenly spaced around it as $n$ increases.